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Actually a further problem- I understand it from the flat sheet to a cone: Cut out a pie from a sheet, Draw a (incomplete) circle around the tip of the pie, Draw arrows pointing in the same direction (parallel) along the circle (i.e. parallel transport on the flat surface with the pie cut out) and now roll the sheet into a cone with the pie tip as cone tip and matching the edges of the pie. The arrows are no longer parallel.

But I have not understood it starting from a cone. Now the surface is curved, what is a parallel vector, does it stay on the surface? How do I try to parallel transport a vector on the circle on a cone? Any clear examples available.

I sort of do understand the Tensor math. I am looking for a visual explanation.